Atemporal Topological Information Dynamics: Emergence of Time via Categorical Amnesia and Non-Hermitian Friction (Part 1/Draft)

To resolve the 'Problem of Time'—a foundational challenge in quantum gravity and open quantum systems—this paper proposes Atemporal Topological Information Dynamics (ATID), a novel information-theoretic framework that operates without a background spacetime manifold. We formulate an effective non-Hermitian Hamiltonian Heff describing open-system dissipation alongside a topological action Sₜₒₚₒ defined over an atemporal information manifold Minfo. Applying the variational principle δ Sₜₒₚₒ = 0, we geometrically derive a non-commutative friction tensor Fabᶠʳⁱᶜ that diverges near Exceptional Points (EPs) due to the non-analytic algebraic structure of the Puiseux series. Furthermore, by introducing a Forgetting Functor Forget mapping from a high-dimensional microscopic topological category TopoSys to a 1D causal order category Time, we prove that thermodynamic entropy and the arrow of time emerge naturally as the 'kernel residue' of lossy topological information compression. Finally, we establish a non-linear algebraic phase delay scaling law (Δ τ ∝ abᶠʳⁱᶜ ln(I_c / (I_c - I))) near the critical information injection threshold I_c, numerically verifying the model via a Parity-Time (PT) symmetric coupled RLC electronic circuit system and providing an accessible experimental protocol.